This tutorial covers material encountered in chapter 9 of the VCE Mathematical Methods Textbook, namely:
- The derivative of functions seen previously in tutorial worksheets 1-5
- Chain, Product and Quotient rules applied to the aforementioned functions
Q1 – Derivatives of Polynomial Functions
Q2 – Gradients of Tangents of Polynomial Functions
Q3 – Derivatives of Exponential, Logarithmic, Sine & Cosine Functions
Q4 – Gradients of Tangents of Exponential, Logarithmic, Sine & Cosine Functions
Q5 – Finding Derivative Zeros
Worksheet
Q1. Find the derivatives of the following with respect to [latex]x[/latex]:
(a) [latex]f(x)=x+\sqrt{2-x^2}[/latex]
(b) [latex]g(x)=\dfrac{x^2-3x+4}{2x^2+1}[/latex]
(c) [latex]h(x)=(x+3)\sqrt{4x+3}[/latex]
(d) [latex]j(x)=\sqrt[3]{5x^2-7}[/latex]
(e) [latex]k(x)=(5x^2-7)^{\frac{1}{3}}[/latex]
Q2. Find the gradient of the tangent of each function at the corresponding points:
(a) [latex]a(x)=4x^2-4x+1,[/latex] at [latex]x=1[/latex]
(b) [latex]b(x)=\dfrac{x-3}{x^4+2},[/latex] at [latex]x=-1[/latex]
(c) [latex]c(x)=\left(2x^2+3 \right)^{\frac{2}{3}}[/latex] at [latex]x=4[/latex]
Q3. Find the derivatives of the following with respect to [latex]x[/latex]:
(a) [latex] f(x)=\ln(x+3)+3 [/latex]
(b) [latex]g(x)=\sin(2x)[/latex]
(c) [latex]h(x)=x^2\cos(3x)[/latex]
(d) [latex]j(x)=\dfrac{\sin(2x+1)}{\cos(2x+1)}[/latex]
(e) [latex]k(x)=e^{2x}\sin(3x)[/latex]
Q4. Find the gradient of the tangent of each function at the corresponding points:
(a) [latex]y(x)=\cos(\pi x)[/latex] at [latex]x=\dfrac{1}{6}[/latex]
(b) [latex]t(x)=xe^{2x},[/latex] at [latex]x=3[/latex]
(c) [latex]r(x)=(4x^2-2)\ln(x-2),[/latex] at [latex]x=3[/latex]
(d) [latex]w(x)= -x\sin(3x),[/latex] at [latex]x=\dfrac{\pi}{4}[/latex]
Q5. For what values of [latex]x\in\R,[/latex] is the derivative of [latex]f(x)=\left(2x+\dfrac{3}{x}\right)^2[/latex] with respect to [latex]x[/latex] zero?
Q6. If the function [latex]f[/latex] is differentiable for all real numbers, find the derivative of each of the following:
(i) [latex]xf(x)[/latex]
(ii) [latex]\dfrac{1}{f(x)}[/latex]
(iii) [latex]\dfrac{x}{f(x)}[/latex]
(iv) [latex]\dfrac{x^2}{\left[f(x)\right]^2}[/latex]
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